Cynthia Lanius

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Table of Contents
 
       Introduction

  Why study fractals?
    What's so hot about
    fractals, anyway?

  Making fractals
    Sierpinski Triangle
         Using Java
         Math questions
         Sierpinski Meets Pascal
    Jurassic Park Fractal
         Using JAVA
         It grows complex
         Real first iteration
         Encoding the fractal
         World's Largest
    Koch Snowflake
         Using Java
         Infinite perimeter
         Finite area
         Anti-Snowflake
            Using Java

  Fractal Properties
    Self-similarity
    Fractional dimension
    Formation by iteration

  For Teachers
    Teachers' Notes
    Teacher-to-Teacher

  Comments
    My fractals mail
    Send fractals mail

  Fractals on the Web
    The Math Forum

  Other Math Lessons
    by Cynthia Lanius

lanius@math.rice.edu

What people on the Internet are saying about my unit:

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From Dr. James E. Shirey, a school board member:

I am a member of two local boards of education, and I am always on the lookout for things that would be useful for talented and gifted programs. This site looks good to me, and I have recommended it to several hundred people in the Ohio Association for Gifted Children.
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From Roland Olstorpe, a Swedish teacher:

In my work with pupils in ninth grade in Sweden where I mix Euclid in modern language with origami work, I have a small part where you can work which in the second generation looks like "The math forum" picture. The tetaedrons are made of two symmetric modules. That makes them steady. The chapter which contains this work is about the latest part of the mathematical history of geometry. Please enter my adress if some one would like to contact me k-r@plinus.se
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From a mathematics college student:

Thank you so much for all this information on fractals. I am a college student majoring in mathematics, but I am planning on being an elementary school teacher when I graduate. Fractals are a wonderful link between what I am learning in classes now and what I can teach students. There definitely needs to be more teacher education out there regarding fractals. I appreciate and thank you for this site.
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From a mathematics professor:

I first encountered the JPF reading Martin Gardner's articles in Scientific American (March 1967, page 216; April 1967, pages 124-125; pages 118-120; July 1967, pages 115 and 217), written well before Mandelbrot coined the term fractal. It occurs to me now that these articles are likely to contain more interesting questions than I posed to you. In case you may be interested there is another "Jurassic Park" fractal called the Terdragon. It also results from paperfolding. This curve was introduced by Davis and Knuth in the Journal of Recreational Mathematics (part I, vol. 3, 1970, pages 61-81; part II, pages 133-149). There, among some technical writing, the authors discuss some lighter themes concerning the JPF and paperfolding. I hope this helps. As you may have guessed, I enjoy playing with, and inventing, spacefilling curves. I find them "SupercalliFractaleerecursiMandelbrotious".
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His earlier mail:

Your web class is beautiful. I hope you inspire others to bring the beauty and challenge of mathematics to a wider audience.

I have played with what you call the Jurassic Park fractal (Heighway's Dragon Curve) and noted some properties your students might enjoy exploring.

  • What if every other line of the Jurassic Park fractal (JPF) is invisible? Will all of the visible lines be horizontal or vertical at all iterations?
  • If the first line of JPF is horizontal for a given iteration, must the last line be horizontal, or must the last line be vertical?
  • The JPF is formed by paperfolding. After the seventh fold, where among all those creases is the very first fold?
All of these problems can be solved by experimentation or analysis of the paperfolding process. One problem I found visually entertaining was to create a rule for which lines should be invisible for JPF. With experimentation the students can discover interesting patterns; for the dull ones they can find logical explanations.
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From a high school student

The area of the Koch Snowflake is (2 sqrt3)/5
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From a student in Singapore

I am a Singaporean student in Secondary 3. I surfed through your site and found it interesting. Thought I might ask just one question on the Jurassic Park fractal. The frac has always been drawn at right angles (each turn in the fractal is 90 degrees either right or left). What would happen to it if the turns were to become 60 degrees or 120 degrees?Hope you could give me an answer soon.
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From a student

I'm another student in Mr. Gross's math class. I think that Jurassic Park fractal is cool. The ones Mr. Gross gave us before we had to do on paper. Now we can do it with the paper! I also read The Lost World(also by Michal Crichton). It that had those iteration-whatsits and I had no idea what that meant. Now I know!! Thanks!!!!
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From another student

I'm from Mr. Gross' math class! I've done your Snowflake, Triangle, and tonight's homework about the dangerous job where you pick to be paid $1,000,000 or one penny a day the first day and gets doubled every day until the 30 days is up. I like all the projects and got the answer of $10,737,418.23 for tonight's homework if you did the penny choice (I hope it's right)! Well, I don't want to take up too much of your time but keep up the good work!!!!
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From a teacher

I recently used your Fractal Page with my eighth grade math class. I used it on a day when the parents were in and they all loved it. We followed your step by step directions and constructed the Sierpinski Triangle. It was a great lesson and my students loved it. The finished triangles decorate the hallway and prompt many questions from other students as well as faculty. Thanks for sharing your ideas.
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From Education World(tm) Site Review:

http://math.rice.edu:80/~lanius/frac/
Fractals
OVERALL:      A
CONTENT:      A       Information on fractals for kids.
AESTHETICS:      A       Simple layout, step-by-step graphic illustrations of projects.
ORGANIZATION:      A+       Sidebar menu appears on each page and shows a check mark on section currently being viewed.

REVIEW:
This page was created specifically for kids because, as the site points out, most of the information on the web about fractals is "either just pretty pictures or very high-level mathematics" This website tries to teach students about having fun with math and incites interest in students about fractals by explaining the whole field of fractals is new and currently being researched and that fractals are more like objects found in nature than circles, squares and triangles (making the idea have a basis in everyday life). The site comes with several projects demonstrating fractals that allow users to work along. Information is available on some famous fractal projects including Sierpinski Triangle, Jurassic Park Fractal and Koch Snowflake and there is also a section for teachers that includes Grade Level suggestions, Lesson Procedures, Mathematics Topics, Materials and Media and a Suggested Assessment for the lesson plans found on the site. Links can also be found for other web resources on fractals. Fractals can seem to be an overwhelming subject and this site promises you'll come away with a better understanding of them, and probably an interest too.

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From a Singapore student:

I am a Singapore student in secondary 3 (9th year) studying in Raffles Institution. I looked through your site (congratulations on being the owner of such a perfect site -- I have never seen one close to it before) and saw the article on the infinite perimeter of fractals. However I looked through it, I am still astonished and amazed by it. Could you send me a fuller explanation of this phenomena?
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From a Pennsylvania student:

Hello. I am a sixth grade student at Snyder Elementary. My teacher, Mr. Gross, taught us how to make the Sierpinski Triangle. I thought it was a very exciting lesson. I got home that night and went to your web site. I really loved the Jurassic Park one. I hope that other teachers find out about the Sierpinski Triangle and your web site.
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From a California teacher:

I am a distance learning instructor from the LA County Office of Educ and our math programs are broadcast nationally... we will be dealing with Pascal and Palindromes (looking for patterns) and thought this was a good time to present "Fractals".
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From a student:

what exactly is the definition of a fractal? im doing a big project about them for geometry and i was wondering if you had any web pages that might have good information about them! thanks!
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From a parent:

My 10 year old daughter Megan and I have just discovered your fractal page as she searched for a possible math fair topic. (Math is a new addition to our sciencefair this year.Last year she designed and presented her own home page.) The information on fractals is new to both of us and I am struggling to keep up with her, however it is quite exciting. Thank you for a well explained lesson. I am still searching for more information which is at a middle school level. Anything you could pass our way would be much appreciated.
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From a teacher turned grad student:

Hi! I ran across your fractals page on the internet and thought you may be interested in a paper that I wrote on iterated function systems and/or a program to plot the attractors of iterated function systems. I taught high school math and technology for 6 years and am currently in grad school in technology education. I have spoken at several Virginia State math conferences on fractals, iterated function systems, and fractal image compression. My info is geared more to the high school/college level, but you may find something useful. Both the paper and program can be found at: http://hamnetcenter.com/jeffc/fractal.html Thanks for sharing your knowledge with the rest of us on the 'net! I will be passing your URL to some of my colleagues who are middle school math teachers. Thanks again!
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From a Connecticut teacher:
Hello, Wow, what a great page. One of our 5th grade teachers is interested in Fractals and I made a link to your page from his classrooms page. I love what your doing with girls in Math also.
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From an Alabama teacher:
I have used your fractal pages in my 5th grade math enrichment class for a student who wanted to do a project on something different. He was fascinated, and successful in following the directions on his own to make the Sierpinski model. His success got others interested! I am hoping to get a whole class model before the year is over. Thanks for this beautiful presentation in language the upper elementary student can understand! I will be making presentations at National Council of Teachers of Mathematics Southern Regional meetings in New Orleans and Atlanta, talking about WWW sites for elementary math enrichment. I would like to share the Fractal pages (using WebWhacker software) and include the URL in my handout. Hoping you can give me permission to share your excellent and informative site.
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Montana Revisited:
We have been busy using the wonderful resources in your fractal page in our studies of fractals this year; I just haven't sent anything back because we meet so infrequently that it takes a long time to complete projects. My students have really enjoyed your web page.
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From a Virginia teacher:
I am a middle grades education major at Averett College in Danville, Virginia. I will be using your fractal lesson this Wednesday at Gibson Middle School here in Danville, so you may be receiving some e-mail with their answers. I would appreciate it if you would allow me to use your lesson on my homepage. I have included your lesson plan on fractals in activities to support our new Standards of Learning. Hope to hear from you soon.
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From a United Kingdom professor:
I have looked at your Fractal lesson pages. They are excellent! I wish I had such a nice introduction to fractals when I was a middle school student! I've already put a link to your page. I appreciate that you referenced my page. Thank you.
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From The Math Forum:
Great pages! I've put the top level page into our Internet resource collection (Steve's Dump) along with two other specific pages (How to make a fractal, Let's Iterate!) and when our script has run early tomorrow morning they'll be in the elementary and middle school levels, graphics and visualization, and math lessons, and searchable by keyword fractal.
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From the Eisenhower National Clearinghouse(ENC):
Your site, http://math.rice.edu:80/~lanius/frac/ A Fractals Lesson has been chosen as one of the ENC-Online's DIGITAL DOZEN for August. Starting August 1st, you may notice some new traffic on your webpages. Attached to this email are two small GIFs that could be displayed on your page, a symbolic badge of appreciation from us to you. Please visit our home, http://www.enc.org , and you can see the link to your page on the DIGITAL DOZEN link there. Thanks for making a great site!"
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From a NASA researcher:
It's great that you're teaching fractal math to young kids! Most of what you have on the Web is pure math. You might also be interested in a site focusing on a scientific application, namely using fractals to make models of different types of clouds: http://climate/~cahalan/FractalClouds/FractalClouds.html Note that our goal isn't to make "pretty cloud pictures", so you won't find many of those at our site. Our goal is to reproduce physical cloud properties, like the spatial spectrum and probability distribution of cloud liquid water, and to compute from these the cloud radiative properties, like the albedo, absorption, and transmission. Cloud radiative properties have a strong impact on surface heating, rainfall, and other aspects of life on planet Earth. Some of the science of Solar and Earth radiation is discussed at: http://climate.gsfc.nasa.gov/~cahalan/Radiation/ Keep up the good work!"
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From an HISD Principal:
"I am impressed with the home pages of the Milby staff. Is this something connected with the school or just the interest of the staff members. I am also with the district. Your page looks very professional. I am sure that it is much more complicated than it looks. You make the page look clean and simple."
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From the Netherlands:
P.S. By the way, did you know that you were mentioned on our "Dialy Planet", a Dutch daily newspaper on the Net? That's how I reached your site. Maybe you're getting famous now :-)!!! Bye!
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From a Baltimore teacher:
I want to thank you for your lesson on fractals for elementary and middle school. I plan to adapt it to the high school geometry courses which I am teaching this year. I was wondering if you have any other lessons aimed specifically at the high school level or if you can recommend any other resources for such. I'm finding it difficult to figure out how to get the students to appreciate just how fascinating and valuable fractals are when the students have not yet reached the level of advanced algebra. It seems every time I start to lead them somewhere wonderful I run into some mathematics that they are not yet ready for. Incidentally, if you are ever in the Baltimore area, check out the third floor of the Science Center in the Inner Harbor. There is an excellent presentation on fractals and related areas such as tesselations and topology.
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From a Montana teacher:
We would like to participate in this project. I have a 4-8 math enrichment group at Helena Flats School in Kalispell, MT. Thanks!
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From a Singapore student:
I study Secondary 1 (equivalent to USA Grade 7) at Raffles Girls' School, Singapore. I would like to participate in your project. Could you please sign me up and then send me a confirmation email? Please reply in English. Thank you very much! Hoping for your favourable reply,

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From an Indiana teacher:
Cynthia- Could you explain the project alittle more. I work with 4th and 5th grade gifted and talented students. I think we might be interested in this project. The internet is something new for my students. They will be anxious to do some different things with the internet. Looking forward to hearing from you real soon.
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From an Australian teacher:
I have got the details re:the fractals for kids project from IECC_Projects. Looks interesting. I teach (Maths and Computers) in Melbourne, Victoria at a government high school. My students range in age from 12 to 17. The Year 7 students (ages 12 - 13) all have laptop computers and I am sure that they would be interested in becoming involved in this project. Are these students older than what you require for project? If not, I would be delighted to receive the details from you about this project.
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From a California teacher:
I really enjoyed "How To Make Fractal". However, could you give me a source for triangular grid paper. Thank you.
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My favorite- a university student:
Hey there, I am a sophomore at The University of Texas. I was just checking out your webpage, and although I have not had enough time to really poke around in it, I wanted to let you know that I think it is terrific! You are doing exactly what I hope to do with my life after I get out of school. I'm really glad there is somebody out there who genuinely loves teaching! Perhaps if you had a little spare time some day, you could share stories/advice with me. In the meantime I should get back to my math homework!
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From an Education Consortia:
I'm XX XXXXX, research associate for the Eisenhower Regional Consortium for Mathematics and Science Education@SERVE (the SouthEast Regional Vision for Education). We are one of the ten regional consortia funded by the US Dept. of Ed. to help teachers, schools, etc. improve in the teaching and learning of mathematics and science. I found your wonderful lesson on fractals from the ENC web site and was hoping if you would allow me to reprint this lesson for our newsletter, the Common Denominator. Every issue we include a lesson plan as a regular feature and in this addition, we were thinking of using your lesson in the context of as an example of lessons that can be accessed through ENC Online, which is another featured story for this volume. Please email me your response ASAP and include your mailing address so we can send you a copy.
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From an Israeli teacher:
Read your pages on fractals, and you are correct they are for grades 4-8, they are really good. I teach gifted girls (even though I am not female) in a special program. We have girls from a region of over 50 miles around, they have a long bus ride to come once a week to the program. If you do start a computer club for girls I am sure that some would like to take part. Just one quick question (not to be a pig) but did you do your home pages by yourself or did you have help (that is did you have a class activity) I hope that you are not insulted by the question.
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From an author:
Is it possible to create a three-dimensional fractal? If so, can it its volume be statistically measurable? A two-dimensional fractal can be created on any sheet of paper and by using a computer. But a three-dimensional fractal, with a physical form, would require some type of material. Clay? Plastic? Putty? We are interested in knowing if this is feasible. Thank yhou. We would also be interested in knowing if fractal creations (in two or three-dimensions) are popular in high schools. This is important to us because it relates to a fictional subject we are trying to create.
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From that same author:
Thank you for your suggestion of using a tretrahedon (pyramid) as a model. This makes sense because it is easily statistically measurable. Is it possible to have the sides of the pyramid ragged and still statistically measurable? What other forms would be suitable for threee-dimensional study? Could a fourteen-year-old boy become intrigued with fractals by studying geometry? Again, out concern is to lay a groundwork for a story that has an unusual approach to characters and their concerns.
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From a New Jersey teacher:
Having taught for 27 years, I am still willing to grow. Your lesson on fractals seems very good for a heterogeneous average math class. I intend to try it soon. Thanks for posting it.
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From an 8th Grade class:
We are the 8th grade Algebra class from St. Isaac Jogues School in St. Clair Shores, Michigan. We enjoyed creating the Sierpinski Triangles. Here are the answers to the fractals questions: 1) 3/4 of the first triangle is NOT shaded. 2) 9/16 of the second triangle is NOT shaded. 3) 27/64 of the third triangle is NOT shaded. 4) 81/256 of the fourth triangle is NOT shaded. 5) If we made the fifth triangle, 243/1024 would NOT be shaded. 6) For any step "n," we could find the fraction that is NOT shaded by using the formula 3 to the n power over 4 to the n power. 7) 243/1024 81/256 27/64 9/16 3/4 As the size of the triangle gets larger, the area that is not shaded gets smaller in comparison to the shaded area. 8) We had trouble finding another pattern.
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From a Math Teacher: The Mandelbrot Set is not a fractal! It doesn't have self-similarity. Check the books.
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From a Mathematician:
Yes, I believe that. In fact, the M-set is the antithesis of a fractal in that every little piece is completely different from every other. On the other hand, the boundary of the M-set (which should be 1 dimensional), is in fact two dimensional. SO in that sense the M-set (or rahter its boundary) is a fractal. But I really believe that a fractal should have some semblance of self-similarity, so that's why I do not think that the M-set is a fractal. The Julia sets, on the other hand, are fractals. Hope that helps.